Section 02
The Idea, Step by Step
Roll a marble across a stretched rubber sheet and it goes straight. Now set a bowling ball in the middle: the sheet sags, and the marble curves toward the dip. Nobody pulled it — it just followed the dent. Einstein's wild idea was that this is gravity. Mass dents space and time, and everything — planets, light, you — simply coasts along the dents. The Sun doesn't reach out and grab the Earth; it shapes the space the Earth is gliding through.
To pin numbers on it, every mass $M$ bends spacetime over a scale called its Schwarzschild radius, $r_s=\dfrac{2GM}{c^2}$. For the Sun that's only about 3 km; for the whole Earth, about 9 mm. Those tiny numbers are why everyday gravity feels so gentle — we live far out on a very shallow part of the dent. The deeper into the dent you sit, the slower your clock ticks: a clock at radius $r$ runs at a rate $\frac{d\tau}{dt}=\sqrt{1-r_s/r}$ compared with a clock far away. This isn't a gadget malfunction — time itself runs slow. Worked example: GPS satellites orbit high up, where the dent is shallower, so their clocks gain about 38 microseconds every day. Your phone quietly corrects for it each time it finds your location.
The precise picture
Formally, gravity is the curvature of four-dimensional spacetime, and free objects follow geodesics — the straightest possible paths through that curved geometry. Einstein's field equations, $G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$, say it in one line: matter and energy ($T_{\mu\nu}$) tell spacetime how to curve ($G_{\mu\nu}$), and that curvature tells matter how to move. The Schwarzschild metric solves these equations around a spherical mass; push $r\to r_s$ and the time-dilation factor $\sqrt{1-r_s/r}\to 0$ — that surface is the event horizon of a black hole, where (in coordinate time) clocks appear to freeze. And when massive bodies accelerate hard — two black holes spiralling together — they shake spacetime itself, radiating gravitational waves whose strain $h\sim10^{-21}$ was first caught by LIGO in 2015.
In the sim, the Mass $M$ slider sets how deep the well is (and therefore $r_s$); radius $r$ picks how far out you sample the time-dilation factor $\Delta t/\Delta\tau$; and the chirp mass $M_c$, frequency $f$, and distance $d$ sliders set the gravitational-wave strain $h$.
Try this in the sim above
First, switch to Schwarzschild mode and raise $M$ — watch $r_s$ and the red event-horizon ring grow with it. Next, slide the radius $r$ down toward $1\,r_s$ and watch $\Delta t/\Delta\tau$ shoot up: clocks effectively stop at the horizon. Finally, in Gravitational Waves mode, push the chirp mass $M_c$ higher (or pull the distance $d$ smaller) and watch the strain readout $h$ climb — bigger, closer mergers ring spacetime louder.
Section 03
Equations & Derivation
Einstein Field Equations
$$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu},\quad G_{\mu\nu}=R_{\mu\nu}-\tfrac{1}{2}Rg_{\mu\nu}$$
Schwarzschild Metric
$$ds^2=-\!\left(1-\frac{r_s}{r}\right)c^2dt^2+\!\left(1-\frac{r_s}{r}\right)^{\!-1}dr^2+r^2d\Omega^2,\quad r_s=\frac{2GM}{c^2}$$
Gravitational Time Dilation & Redshift
$$\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}},\quad z=\frac{\Delta\lambda}{\lambda}=\frac{1}{\sqrt{1-r_s/r}}-1$$
Gravitational Wave Strain
$$h=\frac{4G}{c^4}\frac{\ddot{Q}}{d},\quad h\sim\frac{4G^{5/3}}{c^4}\frac{(M_c\pi f)^{2/3}}{d}\quad\text{(compact binary)}$$
Orbital Precession (GR)
$$\Delta\phi=\frac{6\pi GM}{a(1-e^2)c^2}\text{ per orbit},\quad\text{Mercury: 43 arcsec/century}$$
Key Constants & Scales
| Symbol | Quantity | Unit/Value |
|---|
| $r_s=2GM/c^2$ | Schwarzschild radius | km (Sun: 3 km, Earth: 9 mm) |
| $G=6.674\times10^{-11}$ | Gravitational constant | N m² kg⁻² |
| $c=2.998\times10^8$ | Speed of light | m s⁻¹ |
| $1\,M_\odot=1.989\times10^{30}$ | Solar mass | kg |
| $\text{1 Mpc}=3.086\times10^{22}$ | Megaparsec | m |
1
Equivalence Principle. Einstein's key insight (1907): a uniformly accelerating frame is locally identical to a gravitational field. This implies light must bend in gravity (photons follow curved spacetime geodesics) and clocks run slower in stronger gravity — gravitational time dilation. Verified by Pound-Rebka (1959) to 1% accuracy.
2
Schwarzschild radius. For mass $M$: $r_s=2GM/c^2$. Sun: $r_s=2.95$ km. Earth: $r_s=8.9$ mm. If compressed below $r_s$, gravity prevents escape even for light — a black hole. At the event horizon, coordinate time freezes; proper time continues for infalling observer. Hawking radiation (quantum) slowly evaporates black holes.
3
Gravitational waves. Accelerating masses radiate spacetime ripples. Strain $h=\Delta L/L$ — LIGO detected $h\sim10^{-21}$ from GW150914 (two $\sim30M_\odot$ black holes merging at 410 Mpc). Frequency matches orbital frequency ($\times2$). Energy radiated: $\sim3M_\odot c^2$ in 0.2 s — luminosity exceeded all visible stars combined.
4
Four GR tests. (1) Perihelion precession of Mercury (43 arcsec/century). (2) Light deflection by Sun (1.75 arcsec). (3) Gravitational redshift (Pound-Rebka, GPS). (4) Gravitational waves (LIGO 2015). All confirmed to high precision.
Ref: Misner, Thorne & Wheeler — Gravitation (Princeton UP, 1973); Carroll — Spacetime and Geometry (2004); Abbott et al. — LIGO GW150914 (PRL 116, 061102, 2016).
Section 05
Common Misconceptions
❌ General relativity only matters near black holes and neutron stars.
✅ GR effects are present everywhere but become detectable at high precision. GPS (everyday use) requires GR corrections. The Pound-Rebka experiment measured GR in a 22.6 m building. Gravitational lensing affects telescope observations of galaxies. GR governs the large-scale structure and expansion of the universe through the Friedmann equations.
📖 Misner, Thorne & Wheeler — Gravitation, §38; Clifford Will — Was Einstein Right? (2nd Ed.).
❌ Time runs slower in a strong gravitational field, so clocks near a black hole eventually stop.
✅ Gravitational time dilation: $d\tau/dt=\sqrt{1-r_s/r}$. As $r\to r_s$: $d\tau/dt\to0$ — in coordinate time, an infalling object asymptotically approaches the horizon. But the infalling observer's proper time is finite — they cross the horizon in finite proper time and experience nothing special there (if the black hole is large enough). The "stopping" is a coordinate artifact, not physical.
📖 Misner, Thorne & Wheeler — Gravitation, §31.
❌ Gravitational waves travel at an unknown speed.
✅ Gravitational waves travel at exactly $c$ in GR (verified by GW170817 + GRB 170817A: same event detected in GW and gamma-rays within 1.7 s of each other after 130 million light-years of travel — speed difference $<10^{-15}c$). This rules out many alternative gravity theories that predict different GW speeds.
📖 Abbott et al. (2017), ApJL 848, L13 — Multimessenger observation of neutron star merger.
Misconception research: Scherr et al. (2001) Am. J. Phys.; Pitts & Schieve — Slightly Bending Spacetime.